dBs
100+ Head-Fier
- Joined
- Jan 6, 2009
- Posts
- 471
- Likes
- 12
Last night the idea hit me to write up a thorough investigation of harmonic distortion. I don’t believe that distortion in music and amplifier design is misunderstood, but I think I can add to the level of understanding of the phenomenon for a broad audience using a highly visual guide.
What is harmonic distortion and what does it mean to your music?
Put simply; harmonic distortion is alterations to the original musical signal that are introduced during manipulation and transportation of that signal. These distortions act as imperfections to the sound. Some can be highly offensive, some can even be enjoyed.
Let’s start with the basics, the fundamental frequency. This is the input frequency, the frequency that you want to hear, the music. To simplify the idea, let’s assume that it is a single constant tone like in figure 1. This fundamental frequency was chosen arbitrarily because it looked good for this use (it didn’t oscillate too fast or amplitude too large).
Figure 1: Basic tone sine wave
This tone is seen in the time domain. Meaning the horizontal x-axis represents the passage of time while the vertical y-axis measures amplitude. The frequency can be determined by how much time elapses over the course of one complete cycle up and down. This is the most common way to express this tone, but it is not the only representation. By applying what’s called a Fourier transform to the tones equation, you convert the equation to the frequency domain.
The frequency domain differs from the time domain in that the x-axis now represents the frequency of the oscillation of the tone. Figure 2 shows some examples of various graphs and their corresponding representation in the frequency domain.
Figure 2: Time domain graphs and corresponding frequency domain graphs (image CO bruker-biospin.com)
You will note, that the corresponding frequency domain graph of a perfect tone is a perfect vertical line in the frequency domain. This makes sense as a perfect tone wave has a perfectly consistent frequency and so when shown in the frequency domain will only be seen at that one location. Something to note that is shown here is that the amplitude of the sine wave (how tall the wave crests are in relation to the wave valleys) is the same amplitude of each side of the frequency domain graph. There are two vertical lines seen because the sine wave is repeated, so a negative frequency is indistinguishable from a positive frequency, thus the frequency domain shows that property. You can watch your music in the frequency domain in Foobar2000. When you play with an equalizer, you are in effect dampening or emphasizing specific frequency ranges that are broken down visually in this way. Also note that these are additive. This will be further analyzed later.
Now we will look at a simple visual analog to the concept of harmonics. Figure 3 shows a series of frequencies. If you wish, you can think of them like a guitar string that is supported from either end and plucked to vibrate. If you were to look at the string, it would actually vibrate in one or [more likely] all of these fashions. The farthest left is the fundamental frequency that I spoke of before. This acts as the basis of all the subsequent harmonics that form that you can see to the right of the fundamental. The fundamental frequency and subsequent harmonics, in the case of a guitar string, are determined by the distance between the supports at either end.
Figure 3: Fundamental and higher order harmonics (image CO cnx.org)
What is important to note is that the harmonics that are produced from the fundamental can be counted. Counting the number of waves tells you what harmonic it is in relation to the fundamental. The fundamental is the first order, the next has two waves within the bounds and thus is the second order, etc.. This shows you visually one of the ways that an odd order harmonic differs from an even order harmonic. This doesn’t show much at this point, that will come later.
Now, the music that you listen to via such a guitar string, your amp, or your best friends’ voice, is comprised of many different sound frequencies at one time. This is what you see when you view a song visually in many audio players (Foobar2000 calls it the oscilloscope, it’s more technical equipment name). Figure 4 shows you how different frequencies combine. You can also look at this as though the green graph is the fundamental and the orange and red are higher order harmonics of differing amplitudes (note that the orange is larger than the green, if these truly were fundamental and harmonic, the orange could never be larger than the green). Another good example of this property was seen before in figure 2.
Figure 4: Harmonic addition in the time domain (image CO National Instruments)
If you were to look at the white sum graph in the frequency domain, you would see a vertical line at the frequency of each of the three harmonics, for a total of three positive and three negative. Similar to what you might see in figure 5.
Figure 5: Fourier transform of a guitar’s A string (image CO ????? woops!)
Here you can see the fundamental at just above 200Hz. You can also see the decaying harmonics that follow. The first spike at just above 300Hz is the second order, the next spike is the third order, etc.. How tall these spikes are is their amplitude. Naturally, the taller the worse. Also, as I will discuss later, the higher the order, the worse.
So, all this is well and good, but what does it all MEAN? I’m glad you asked (and maybe even still reading…HA!).
There are different types of repeating graphs; sine wave, cosine wave, triangle wave, square wave, sawtooth wave, etc.. The ones we are interested in right now is the square and triangle wave. These are important because these represent an infinitely decaying order of odd (square wave) and even (triangle wave) order harmonics. Fourier theorized that all complex waves are comprised of other fundamental waves added together. The equation we are interested in (for odd order) is
. What is most important to take from this is that big “E” looking thing, known as a sigma. This is a series of terms that are added together (like in figure 4 above with the red, orange and green waves). The number of terms goes to infinity (in this case only the odd harmonics; 1, 3, 5,…). In the real world, this does not happen, but thinking of them going to infinity illustrates a VERY pertinent differentiating factor of even versus odd order harmonics.
When odd order harmonics are summed to infinity, they form a perfect square wave. By contrast, when even order harmonics are summed to infinity, they form a triangle wave. By looking at figures 6 and 7 we can this occurrence well. What is important to note is how significantly different the square wave looks from the fundamental (the animations starting frame) than the triangle wave. The triangle wave only varies slightly from the fundamental, desired, sound. This is why odd order harmonics sound so much more offensive than even order harmonics. Odd order harmonics differ more from the sound of the music than does the even order. This also explains why the higher the order, the more offending the harmonic, very high order harmonics at significantly smaller amplitudes can be much more offending than third order harmonic at a much larger amplitude. The estute will notice that the places where the odd and even order change the least is at the node points.
Figure 6: Odd order harmonics summing (image CO wiki)
Figure 7: Even order harmonics summing (image CO wiki)
Now, let’s look at the difference between tube harmonic distortion and solid state harmonic distortion. These two amplify on similar principles but using different portions of their characteristic curves (I won’t go into characteristic curve details here as I don’t want to get distracted from my focus of harmonics. There are plenty of very good explanations about the characteristic curves and their significance online). Figure 8 shows a sample set of solid state characteristic curves for a solid state device (not all mind you, just one type). Figure 9 shows the same for a triode. An amp operates by fluctuating the gate (ss) or grid (tube), effectively sliding along the resistive load line left and right (though also up and down).
Figure 8: Solid state characteristic curves (image CO faqs.org)
Figure 9: Triode characteristic curves (image CO turneraudio.com)
The more “linear” a devices’ curves are, the less distortion the device will color the music with. Linear is a way to say evenly spaced curve lines along the load line. Imagine a signal coming in and forcing the device to slide across part of the load line. If the curve lines are far spaced at one end of the swing and close spaced at the other end, you would get a lopsided output sound. Imagine a sine wave that’s been squished on the negative or positive part of the curve. There is no perfect device, so this “squishing” makes itself known in the form of harmonic distortion. Tubes happen to favor even order distortion in their operation while solid state devices favor odd order harmonic distortion (though at significantly lower levels).
Now we get to the final important point, cut off. I consider this a smaller issue as I don’t tend to push my headphones hard enough to reach cut off. Cut off is when you turn the gain through the device up too high and force the swing on the load line to hit values that literally cut off. If you want to know more of the technical’s on this, there are very good explanations online. Cut off creates VERY bad harmonic distortions of the worst kind in SS devices; odd high order. This is because cut off in a SS device is a hard limit that creates a flat top or bottom to the curves (much like the square wave), an affect you can see in figure 10. Alternatively, tubes are more flexible when they reach cut off. They have a tendency to taper instead of hard cut. This allows for the harmonics to be less impactful as the curve is still relatively maintained, as can be seen in figure 11.
Figure 10: Hard cut off on a SS device. Note similarity to square wave (image CO geofex.com)
Figure 11: Soft cut off on a tube device. Note the lack of hard cut off (image CO geofex.com)
Some talk of the “tube sound”. This is often actually an emphasis of the even order harmonic distortions. Studies have shown that high order odd harmonic distortions are extremely offensive to the ear and small amounts can be easily distinguished from the music.
I won’t get into the transfer function aspects of harmonic distortion as Pete Millet presents a very good explanation of their relationship here (as well as some alternative explanations of what I present here): http://www.pmillett.com/file_downloads/ThesoundofDistortion.pdf
Well, there you go. Probably more than anyone ever wanted to know about this TERRIBLY interesting topic XD. I wanted this to be readable to anyone who has a basic math knowledge so I apologize if it seemed like I was patronizing, as that was not my intent. I will hopefully be adding to this as I still feel it is incomplete, especially at the end when you can tell I was getting burnt out, lol.
I hope this helped make things clearer for anyone who had any confusion on the subject matter. If anyone would like any clarification on anything or any further information, I will do my best to help. This can become a VERY complex topic though with many PhD theses in this field, but I will do what I can.
What is harmonic distortion and what does it mean to your music?
Put simply; harmonic distortion is alterations to the original musical signal that are introduced during manipulation and transportation of that signal. These distortions act as imperfections to the sound. Some can be highly offensive, some can even be enjoyed.
Let’s start with the basics, the fundamental frequency. This is the input frequency, the frequency that you want to hear, the music. To simplify the idea, let’s assume that it is a single constant tone like in figure 1. This fundamental frequency was chosen arbitrarily because it looked good for this use (it didn’t oscillate too fast or amplitude too large).
Figure 1: Basic tone sine wave
This tone is seen in the time domain. Meaning the horizontal x-axis represents the passage of time while the vertical y-axis measures amplitude. The frequency can be determined by how much time elapses over the course of one complete cycle up and down. This is the most common way to express this tone, but it is not the only representation. By applying what’s called a Fourier transform to the tones equation, you convert the equation to the frequency domain.
The frequency domain differs from the time domain in that the x-axis now represents the frequency of the oscillation of the tone. Figure 2 shows some examples of various graphs and their corresponding representation in the frequency domain.
Figure 2: Time domain graphs and corresponding frequency domain graphs (image CO bruker-biospin.com)
You will note, that the corresponding frequency domain graph of a perfect tone is a perfect vertical line in the frequency domain. This makes sense as a perfect tone wave has a perfectly consistent frequency and so when shown in the frequency domain will only be seen at that one location. Something to note that is shown here is that the amplitude of the sine wave (how tall the wave crests are in relation to the wave valleys) is the same amplitude of each side of the frequency domain graph. There are two vertical lines seen because the sine wave is repeated, so a negative frequency is indistinguishable from a positive frequency, thus the frequency domain shows that property. You can watch your music in the frequency domain in Foobar2000. When you play with an equalizer, you are in effect dampening or emphasizing specific frequency ranges that are broken down visually in this way. Also note that these are additive. This will be further analyzed later.
Now we will look at a simple visual analog to the concept of harmonics. Figure 3 shows a series of frequencies. If you wish, you can think of them like a guitar string that is supported from either end and plucked to vibrate. If you were to look at the string, it would actually vibrate in one or [more likely] all of these fashions. The farthest left is the fundamental frequency that I spoke of before. This acts as the basis of all the subsequent harmonics that form that you can see to the right of the fundamental. The fundamental frequency and subsequent harmonics, in the case of a guitar string, are determined by the distance between the supports at either end.
Figure 3: Fundamental and higher order harmonics (image CO cnx.org)
What is important to note is that the harmonics that are produced from the fundamental can be counted. Counting the number of waves tells you what harmonic it is in relation to the fundamental. The fundamental is the first order, the next has two waves within the bounds and thus is the second order, etc.. This shows you visually one of the ways that an odd order harmonic differs from an even order harmonic. This doesn’t show much at this point, that will come later.
Now, the music that you listen to via such a guitar string, your amp, or your best friends’ voice, is comprised of many different sound frequencies at one time. This is what you see when you view a song visually in many audio players (Foobar2000 calls it the oscilloscope, it’s more technical equipment name). Figure 4 shows you how different frequencies combine. You can also look at this as though the green graph is the fundamental and the orange and red are higher order harmonics of differing amplitudes (note that the orange is larger than the green, if these truly were fundamental and harmonic, the orange could never be larger than the green). Another good example of this property was seen before in figure 2.
Figure 4: Harmonic addition in the time domain (image CO National Instruments)
If you were to look at the white sum graph in the frequency domain, you would see a vertical line at the frequency of each of the three harmonics, for a total of three positive and three negative. Similar to what you might see in figure 5.
Figure 5: Fourier transform of a guitar’s A string (image CO ????? woops!)
Here you can see the fundamental at just above 200Hz. You can also see the decaying harmonics that follow. The first spike at just above 300Hz is the second order, the next spike is the third order, etc.. How tall these spikes are is their amplitude. Naturally, the taller the worse. Also, as I will discuss later, the higher the order, the worse.
So, all this is well and good, but what does it all MEAN? I’m glad you asked (and maybe even still reading…HA!).
There are different types of repeating graphs; sine wave, cosine wave, triangle wave, square wave, sawtooth wave, etc.. The ones we are interested in right now is the square and triangle wave. These are important because these represent an infinitely decaying order of odd (square wave) and even (triangle wave) order harmonics. Fourier theorized that all complex waves are comprised of other fundamental waves added together. The equation we are interested in (for odd order) is
When odd order harmonics are summed to infinity, they form a perfect square wave. By contrast, when even order harmonics are summed to infinity, they form a triangle wave. By looking at figures 6 and 7 we can this occurrence well. What is important to note is how significantly different the square wave looks from the fundamental (the animations starting frame) than the triangle wave. The triangle wave only varies slightly from the fundamental, desired, sound. This is why odd order harmonics sound so much more offensive than even order harmonics. Odd order harmonics differ more from the sound of the music than does the even order. This also explains why the higher the order, the more offending the harmonic, very high order harmonics at significantly smaller amplitudes can be much more offending than third order harmonic at a much larger amplitude. The estute will notice that the places where the odd and even order change the least is at the node points.
Figure 6: Odd order harmonics summing (image CO wiki)
Figure 7: Even order harmonics summing (image CO wiki)
Now, let’s look at the difference between tube harmonic distortion and solid state harmonic distortion. These two amplify on similar principles but using different portions of their characteristic curves (I won’t go into characteristic curve details here as I don’t want to get distracted from my focus of harmonics. There are plenty of very good explanations about the characteristic curves and their significance online). Figure 8 shows a sample set of solid state characteristic curves for a solid state device (not all mind you, just one type). Figure 9 shows the same for a triode. An amp operates by fluctuating the gate (ss) or grid (tube), effectively sliding along the resistive load line left and right (though also up and down).
Figure 8: Solid state characteristic curves (image CO faqs.org)
Figure 9: Triode characteristic curves (image CO turneraudio.com)
The more “linear” a devices’ curves are, the less distortion the device will color the music with. Linear is a way to say evenly spaced curve lines along the load line. Imagine a signal coming in and forcing the device to slide across part of the load line. If the curve lines are far spaced at one end of the swing and close spaced at the other end, you would get a lopsided output sound. Imagine a sine wave that’s been squished on the negative or positive part of the curve. There is no perfect device, so this “squishing” makes itself known in the form of harmonic distortion. Tubes happen to favor even order distortion in their operation while solid state devices favor odd order harmonic distortion (though at significantly lower levels).
Now we get to the final important point, cut off. I consider this a smaller issue as I don’t tend to push my headphones hard enough to reach cut off. Cut off is when you turn the gain through the device up too high and force the swing on the load line to hit values that literally cut off. If you want to know more of the technical’s on this, there are very good explanations online. Cut off creates VERY bad harmonic distortions of the worst kind in SS devices; odd high order. This is because cut off in a SS device is a hard limit that creates a flat top or bottom to the curves (much like the square wave), an affect you can see in figure 10. Alternatively, tubes are more flexible when they reach cut off. They have a tendency to taper instead of hard cut. This allows for the harmonics to be less impactful as the curve is still relatively maintained, as can be seen in figure 11.
Figure 10: Hard cut off on a SS device. Note similarity to square wave (image CO geofex.com)
Figure 11: Soft cut off on a tube device. Note the lack of hard cut off (image CO geofex.com)
Some talk of the “tube sound”. This is often actually an emphasis of the even order harmonic distortions. Studies have shown that high order odd harmonic distortions are extremely offensive to the ear and small amounts can be easily distinguished from the music.
I won’t get into the transfer function aspects of harmonic distortion as Pete Millet presents a very good explanation of their relationship here (as well as some alternative explanations of what I present here): http://www.pmillett.com/file_downloads/ThesoundofDistortion.pdf
Well, there you go. Probably more than anyone ever wanted to know about this TERRIBLY interesting topic XD. I wanted this to be readable to anyone who has a basic math knowledge so I apologize if it seemed like I was patronizing, as that was not my intent. I will hopefully be adding to this as I still feel it is incomplete, especially at the end when you can tell I was getting burnt out, lol.
I hope this helped make things clearer for anyone who had any confusion on the subject matter. If anyone would like any clarification on anything or any further information, I will do my best to help. This can become a VERY complex topic though with many PhD theses in this field, but I will do what I can.
